Articles of fourier transform

Using Parseval's theorem to solve an integral

The question at hand is to use Parseval’s theorem to solve the following integral: $$\int_{-\infty}^{\infty} sinc^4 (kt) dt$$ I understand Parseval’s theorem to be: $$E_g = \int_{-\infty}^{\infty} g^2(t) = \int_{-\infty}^{\infty} |G(f)|^2 df $$ I began by doing the obvious and removing the squared such that: $$g^2(t) = Sinc^4 (kt)$$ $$g(t) = Sinc^2 (kt)$$ Following the […]

Computing Fourier transform of power law

I’m trying to compute the Fourier transform of $$f(\mathbf{r}) = \frac{1}{r^\alpha}$$ where $\mathbf{r} \in \mathbb{R}^n$. For sufficiently large $\alpha$, the Fourier transform exists. One well-known example in physics is the case $\alpha = n-2$, which is the Coulomb potential; its Fourier transform is $1/k^2$. For a general $\alpha$, I can argue that $$\widetilde{f}(\mathbf{k}) \sim \frac{1}{k^{n-\alpha}}$$ […]

Find the solution of the Dirichlet problem in the half-plane y>0.

Find the solution of the Dirichlet problem in the half-plane $y>0$. $${u_y}_y +{u_x}_x=0, -\infty<x<\infty,y>0$$ $$u(x,0)=f(x),-\infty<x<\infty$$ $u$ and $u_x$ vanish as $$ \lvert x\rvert\rightarrow\infty$$ and $u$ is bounded as $$y\rightarrow\infty$$ I’ve tried hard for this,but my answer is not matching.I’ve gone through by applying infinite fourier transformation as my text book suggests,but I’m not even getting […]

Is Plancherel's theorem true for tempered distribution?

Let $f, g\in L^{2},$ by Plancherel’s theorem, we have $$\langle f, g \rangle= \langle \hat{f}, \hat{g} \rangle.$$ My Question is: Is it true that: $$\langle f, g \rangle= \langle \hat{f}, \hat{g} \rangle$$ for $f\in \mathcal{S’}(\mathbb R^d)$ (tempered distribution) and $g\in \mathcal{S}(\mathbb R^d)$(Schwartz space)? If yes, how to justify.

Does this integral have a closed form or asymptotic expansion? $\int_0^\infty \frac{\sin(\beta u)}{1+u^\alpha} du$

I’m interested in this following definite integral: $$ \int_0^\infty du \frac{\sin(\beta u)}{1+u^\alpha}, $$ where $\beta>0$ and $\alpha\geq1$. Is there any closed form for this integral? I would be fine if the answer involves special functions, it would just be nice to have the answer in closed form. I’m also interested in the behavior of this […]

Fourier Transform of $\exp{(A\sin(x))}$

I would like to know how to calculate the Fourier transform of $$e^{A\sin(x)}$$ where $A$ is a real positive constant.

Fourier Transform Proof $ \mathcal F(f(x)g(x))=(\frac{1}{2\pi})F(s)*G(s)$

I need to prove this: $$ \mathcal F(f(x)g(x))=(\frac{1}{2\pi})F(s)*G(s)$$ So far, I believe I have to use the Fourier transform standard equation $$ \mathcal F(f(x))=\frac{1}{2\pi}\int_{-\infty}^\infty f(x)e^{-isx}\,dx $$

find the fourier cosine transform of the function defined by $\displaystyle f(x)= \frac1{1+x^2}$

Don’t know how to answer this question. please give the answer if anyone knows.. I am using this equation \begin{align} \sqrt{\frac{2}{\pi}}\int^\infty_0 f(x) \cos (\omega x) \,dx \end{align}

Dirac Delta and Exponential integral

I am able to derive the following equation by substituting the definition of a Fourier transform into it’s inverse. $$2\pi\delta(x-x’) = \int_{-\infty}^{\infty} e^{ik(x-x’)} dk$$ How do you prove that the Dirac Delta is equal to an integral of the exponential function? How do you prove the above equation is true?

How to prove this equality .

Let $f(z)=(\tau+z)^{-k}$. I want to apply the Poisson summation formula, to get that : $$\sum_{n \in \mathbb{Z}} (\tau+n)^{-k} = \frac{(-2 \pi i)^{k}}{(k-1)!} \sum_{m=1}^{\infty} m^{k-1}e^{2 \pi i m \tau}. $$ But then how can I compute the Fourier transform? I was thinking in residues but I don’t know which contour will be easiest for the computation, […]